A ticket-system like JIRA is the same as an e-mailclient to me. You sort by activity, having the most active tickets on top. You can flag/mark them, you can put them in different boxes or just use tags,... All are just means to create order in chaos.
But I would never think to move my old e-mail to a different 'system' than my normal e-mail(client). I never think of old e-mails or drafts as something I need to clean up.
I'll frequently have an idea and toss it into a backlog ticket. Maybe just as a title to start. When a customer mentions that problem/feature, I can add that info in the backlog even if it doesn't immediately become a priority.
When the issue does become something worth working on, I'll usually add more details but the various evidence accumulated over 6 months of letting the item "bake" is invaluable.